VOIGT MODEL Maxwell mdel essentially assumes a uniform distribution Of stress.Now assume uniform distribution of strain VOIGT MODEL
Picture representation Equation
__ σ(t) = Eε(t) +η dε(t) dt (Strain in both elements of the model is the same and the total stress is the sum of the two contributions)
VOIGT MODEL - creep and stress relaxation Gives a retarded elastic response but does not allow for “ideal” stress relaxation,in that the model cannot be “instantaneously” deformed to a given strain.
But in CREEP σ = constant,σ __ σ(t) = σ 0 = Eε(t) +η dε(t) dt Strain
σ dε(t) __ + ε(t) __ _0 η dt τ t‘ σ 0 ε(t) = _ τ t‘ )] Ε [1- exp (-t/ =
t
1
t2
t
τ t‘ - retardation
time (η/E)
0
SUMMARY
E
E E
η
η
Spring
Maxwell element
Dashpot
Spring
Strain
t1
t2
Voigt element
Maxwell model
Dashpot
t1
η
t2
t1
t2
Voigt model
t1
t2
t
PROBLEMS WITH SIMPLE MODELS •The maxwell model cannot account for a retarded elastic response •The voigt model does not describe stress relaxation •Both models are characterized by single relaxation times - a spectrum of relaxation times would provide a better description NEXT - CONSIDER THE FIRST TWO PROBLEMS THEN -THE PROBLEM OF A SPECTRUM OF RELAXATION TIMES
FOUR - PARAMETER MODEL ELASTIC + VISCOUS FLOW + RETARDED ELASTIC eg CREEP
E
σ _0 +σ __ _ 0 [1- exp (-t/ 0t ε =σ + τ t )] ΕM ηM ΕM
M
η
EV
V
η M
Strain Retarded elastic response Elastic response t1 applied
Permanent deformation t2 removed
Time (t)
DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES The Maxwell - Wiechert Model 1 dσ dε _ = σ _1+_ _1 η 1 Ε 1dt dt σ 1 dσ _2 2 _ = _ + η 2 Ε 2dt σ 1 dσ _3 3 _ = _ + η 3 Ε 3dt Consider stress relaxation
E1
η1
dε _ =0 dt
σ 1 = σ 0 exp[-t/τ t1 ] σ 2 = σ 0 exp[-t/τ t2 ] σ 3 = σ 0 exp[-t/τ t3 ]
E2
η2
E3
η3
DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES Stress relaxation modulus
E(t) = σ(t)/ε 0
E1
η1
σ(t) = σ 1 +σ 2 +σ 3 σ 01 exp (-t/ σ 02 __ E(t) =__ τ ) + 01 ε0 ε0
Or, in general
E2
η2
σ 03 exp (-t/ τ02 ) +__ ε
E(t) = Σ E n exp (-t/τtn ) where
0
E3
η3
exp (-t/ τ03 )
σ 0n __ En = ε 0
SIMILARLY, FOR CREEP COMPLIANCE COMBINE VOIGT ELEMENTS TO OBTAIN
τ tn‘ )] D(t) =ΣDn [1- exp (-t/
DISTRIBUTIONS OF RELAXATION AND RETARDATION TIMES Example - The Maxwell - Wiechert Model with n = E(t) = Σ E n exp (-t/τtn )
10
n = 2 Log E(t) (Pa)
8 Glassy region Transition
4
2
Log E (t) (dynes cm- )
10
6
Rubbery plateau
8
2
0 -2
6
4 - 10
Low molecular weight -8
High molecular weight -6
-2 -4 Log time (sec)
0
+2
-1
0
1 2 Log time (min)
3
4
TIME - TEMPERATURE SUPERPOSITION PRINCIPLE Recall that we have seen that there is a time - temperature equivalence in behaviour Glassy region
Log E (Pa)
Glassy region
Transition
10 2
Log E (t) (dynes cm- )
10 9 8 7 Cross-linked elastomers
6 5 4
8
6
Rubbery plateau
4
Low molecular weight
- 10 Melt
Rubbery plateau
Low molecular weight -8
High molecular weight -6
-2 -4 Log time (sec)
0
+2
3 Temperature
This can be expressed formally in terms of a suprposition principle
TIME TEMPERATURE SUPERPOSITION PRINCIPLE - creep T3 > T2 > T1 ε(t) T σ0
T3
T2
T1
Log t
log aT
ε(t) T σ0
0
T0 log t - log aT
T
TIME TEMPERATURE SUPERPOSITION PRINCIPLE - stress relaxation
SIGNIFICANCE OF SHIFT FACTOR What is the significance of the log scale forT a ,and what does this tell us about the temperature dependence of relaxation behaviour in amorphous polymers ? Consider stress relaxation:
E(t) = Σ E n exp (-t/τ t ) n
Let a particular mode of relaxation have a characteristic time τ t0 at T τ t1 at T DEFINE 0 , and a characteristic time 1 . Then
aT
τ t1 = __ τ t0
So that the exponential term can be written
t t __ ___ τ t1 = aT τ t0
Hence, taking logs
log (t/τ t1 ) = log (t/ τt0 ) + log Ta
SIGNIFICANCE OF SHIFT FACTOR log (t/τ t1 ) = log (t/ τt0 ) + log Ta •ie relaxation behaviour at one temperature can be superimposed on that at another by shifting an amount aT along a log scale. •BUT ,real behaviour is characterized by a distribution of relaxation times and relaxation mechanisms vary and have different length scales as a function of temperature •This implies that all the relaxation processes involved have (more or less) the same temperature dependence
RELAXATION PROCESSES ABOVE Tg - THE WLF EQUATION From empirical observation -C1 (T -s T ) _________ Log aT = For Tg > T < ~Tg + 0100 C C2 + (T -s T ) Originally thought that C1 and2 C were universal constants, = 17.44 and 51.6, respectively,when T = Tg .Now known s that these vary from polymer to polymer. Homework problem - show how the WLF equation can be obtained from the relationship of viscosity to free volume as expressed in the Doolittle equation
DYNAMICS OF POLYMER CHAINS An advanced topic that we will not discuss in detail
Rouse - Bouche model A chain as a string of Beads linked by springs
chain
Reptation,scaling concepts And other advanced theories obstacles